Broth

Bring chicken broth, sliced shallots, and mushroom stems to boil in large saucepan. Reduce heat to medium and simmer until mixture is reduced to 3 cups, 35 to 40 minutes. Strain. Return broth to saucepan; discard solids in strainer. DO AHEAD Can be made 1 day ahead. Cover and chill. Rewarm before continuing.

Line rimmed baking sheet with parchment paper. Spoon some flour onto large plate. Working in batches, drop heaping teaspoonfuls gnudi dough (size of small walnuts) onto plate with flour to form about 36 gnudi. Using floured hands, gently shape each into 1 1/2-inch-long, 1/2-inch-thick oval. Tap off excess flour; transfer gnudi to prepared baking sheet. DO AHEAD Can be made 2 hours ahead. Cover and chill.

Bring large wide pot of salted water to boil. Slide gnudi into pot; cook until tender, stirring occasionally, about 10 minutes.